- #1

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## Homework Statement

A is a 2 x 2 matrix. Prove that the set W = {X: XA = AX} is a subspace of M

_{2,2}

## Homework Equations

## The Attempt at a Solution

I have already proven non-emptiness and vector addition.

Non-emptiness:

Code:

```
W must be non-empty because the identity matrix I is an element of W.
IA = AI
A = A
```

Code:

```
Let X, Y be elements of W such that XA = AX, YA = AY. Add the equations together.
(X+Y)A = A(X+Y)
XA+YA = AX+AY
Hence, X+Y is an element of W.
```

Code:

```
Let X be an element of W, and c be a scalar.
cXA = AcX
(cX)A = A(cX)
```

Code:

```
Let X be an element of W, and c be a scalar.
XA = AX
c(XA) = c(AX)
```

I don't understand how to prove that cX is actually in W. This doesn't make any sense to me. How do I prove that?